Introduction Compressible vortex sheets Main result Related problems 2D compressible vortex sheets Paolo Secchi Department of Mathematics Brescia University Joint work with J.F. Coulombel EVEQ 2008, International Summer School on Evolution Equations, Prague, Czech Republic, 16 - 20. 6. 2008 P. Secchi (Brescia University) Compressible vortex sheets
Introduction Euler’s equations of isentropic gas dynamics Compressible vortex sheets Smooth and piecewise smooth solutions Main result Existence results Related problems Plan 1 Introduction Euler’s equations of isentropic gas dynamics Smooth and piecewise smooth solutions Existence results 2 Compressible vortex sheets Compressible vortex sheets Linear Spectral Stability Formulation of the problem 3 Main result Linear stability: L 2 estimate Linear stability: Tame estimate in Sobolev norm Nonlinear stability: Nash-Moser iteration 4 Related problems Weakly stable shock waves Subsonic phase transitions P. Secchi (Brescia University) Compressible vortex sheets
Introduction Euler’s equations of isentropic gas dynamics Compressible vortex sheets Smooth and piecewise smooth solutions Main result Existence results Related problems Euler’s equations of isentropic gas dynamics We consider a compressible inviscid fluid described by the density ρ ( t, x ) ∈ R the velocity field u ( t, x ) ∈ R d the pressure p = p ( ρ ) , where p ∈ C ∞ , p ′ > 0 , whose evolution is governed by the Euler equations � ∂ t ρ + ∇ x · ( ρ u ) = 0 , (1) ∂ t ( ρ u ) + ∇ x · ( ρ u ⊗ u ) + ∇ x p ( ρ ) = 0 , where t ≥ 0 denotes the time variable, x ∈ R d the space variable. P. Secchi (Brescia University) Compressible vortex sheets
Introduction Euler’s equations of isentropic gas dynamics Compressible vortex sheets Smooth and piecewise smooth solutions Main result Existence results Related problems Plan 1 Introduction Euler’s equations of isentropic gas dynamics Smooth and piecewise smooth solutions Existence results 2 Compressible vortex sheets Compressible vortex sheets Linear Spectral Stability Formulation of the problem 3 Main result Linear stability: L 2 estimate Linear stability: Tame estimate in Sobolev norm Nonlinear stability: Nash-Moser iteration 4 Related problems Weakly stable shock waves Subsonic phase transitions P. Secchi (Brescia University) Compressible vortex sheets
Introduction Euler’s equations of isentropic gas dynamics Compressible vortex sheets Smooth and piecewise smooth solutions Main result Existence results Related problems Smooth solutions The Euler equations may be written as a symmetric hyperbolic system. This allows to solve locally in time the Cauchy problem: Initial data ρ 0 ∈ ρ + H s ( R d ) , u 0 ∈ H s ( R d ) with s > 1 + d/ 2 . Existence and uniqueness of a solution in the space C ([0 , T ]; ρ + H s ( R d )) × C ([0 , T ]; H s ( R d )) [Kato, 1975] Finite time blow-up of smooth solutions [Sideris, 1985] Formation of singularities (shock waves). Global (in time) smooth solutions [Serre, 1997] [Grassin, 1998] Local smooth solution of the initial boundary value problem under the slip boundary condition u · ν = 0 (characteristic boundary) [Beirao da Veiga, 1981] P. Secchi (Brescia University) Compressible vortex sheets
Introduction Euler’s equations of isentropic gas dynamics Compressible vortex sheets Smooth and piecewise smooth solutions Main result Existence results Related problems Piecewise smooth solutions The function � ( ρ + , u + ) if x d > ϕ ( t, x 1 , . . . , x d − 1 ) ( ρ, u ) := ( ρ − , u − ) if x d < ϕ ( t, x 1 , . . . , x d − 1 ) , is a weak solution of the Euler equations if ( ρ ± , u ± ) is a smooth solution on either sides of the interface Σ := { x d = ϕ ( t, x 1 , . . . , x d − 1 ) } and it satisfies the Rankine-Hugoniot jump conditions at Σ : ∂ t ϕ [ ρ ] − [ ρ u · ν ] = 0 , (2) ∂ t ϕ [ ρ u ] − [( ρ u · ν ) u ] − [ p ] ν = 0 , ν is a (space) normal vector to Σ ; [ q ] := q + − q − denotes the jump of q across Σ . Σ is an unknown of the problem. Free boundary problem ! P. Secchi (Brescia University) Compressible vortex sheets
Introduction Euler’s equations of isentropic gas dynamics Compressible vortex sheets Smooth and piecewise smooth solutions Main result Existence results Related problems Plan 1 Introduction Euler’s equations of isentropic gas dynamics Smooth and piecewise smooth solutions Existence results 2 Compressible vortex sheets Compressible vortex sheets Linear Spectral Stability Formulation of the problem 3 Main result Linear stability: L 2 estimate Linear stability: Tame estimate in Sobolev norm Nonlinear stability: Nash-Moser iteration 4 Related problems Weakly stable shock waves Subsonic phase transitions P. Secchi (Brescia University) Compressible vortex sheets
Introduction Euler’s equations of isentropic gas dynamics Compressible vortex sheets Smooth and piecewise smooth solutions Main result Existence results Related problems Existence results Existence of one uniformly stable shock wave [Blokhin, 1981] [Majda,1983] Existence of two uniformly stable shock waves [M´ etivier, 1986] Existence of one rarefaction wave [Alinhac, 1989] Existence of sound waves [M´ etivier, 1991] Existence of one small shock wave [Francheteau & M´ etivier, 2000] P. Secchi (Brescia University) Compressible vortex sheets
Introduction Compressible vortex sheets Compressible vortex sheets Linear Spectral Stability Main result Formulation of the problem Related problems Plan 1 Introduction Euler’s equations of isentropic gas dynamics Smooth and piecewise smooth solutions Existence results 2 Compressible vortex sheets Compressible vortex sheets Linear Spectral Stability Formulation of the problem 3 Main result Linear stability: L 2 estimate Linear stability: Tame estimate in Sobolev norm Nonlinear stability: Nash-Moser iteration 4 Related problems Weakly stable shock waves Subsonic phase transitions P. Secchi (Brescia University) Compressible vortex sheets
Introduction Compressible vortex sheets Compressible vortex sheets Linear Spectral Stability Main result Formulation of the problem Related problems Compressible vortex sheets ( ρ, u ) is a contact discontinuity if the Rankine-Hugoniot conditions ( ?? ) are satisfied in the form ∂ t ϕ = u + · ν = u − · ν , p + = p − . p monotone gives equivalently ∂ t ϕ = u + · ν = u − · ν , ρ + = ρ − . The front Σ := { x d = ϕ ( t, x 1 , . . . , x d − 1 ) } is characteristic with respect to either side. Density and normal velocity are continuous across the front Σ . Jump of tangential velocity ⇒⇒ vortex sheet. P. Secchi (Brescia University) Compressible vortex sheets
Introduction Compressible vortex sheets Compressible vortex sheets Linear Spectral Stability Main result Formulation of the problem Related problems We want to show the (local) existence of compressible vortex sheets (contact discontinuities). P. Secchi (Brescia University) Compressible vortex sheets
Introduction Compressible vortex sheets Compressible vortex sheets Linear Spectral Stability Main result Formulation of the problem Related problems Plan 1 Introduction Euler’s equations of isentropic gas dynamics Smooth and piecewise smooth solutions Existence results 2 Compressible vortex sheets Compressible vortex sheets Linear Spectral Stability Formulation of the problem 3 Main result Linear stability: L 2 estimate Linear stability: Tame estimate in Sobolev norm Nonlinear stability: Nash-Moser iteration 4 Related problems Weakly stable shock waves Subsonic phase transitions P. Secchi (Brescia University) Compressible vortex sheets
Introduction Compressible vortex sheets Compressible vortex sheets Linear Spectral Stability Main result Formulation of the problem Related problems Linear Spectral Stability (Landau, Miles, . . . ) Linearize the Euler equations around a piecewise constant vortex sheet � ( ρ, v, 0) , if x d > 0 , ( ρ, u ) = ( ρ, − v, 0) , if x d < 0 . If d = 3 , the linearized equations do not satisfy the Lopatinskii condition ( ∃ exponentially exploding modes!) ⇒ violent instability. √ If d = 2 , and | [ u · τ ] | < 2 2 c ( ρ ) the linearized equations do not satisfy the Lopatinskii condition ⇒ violent instability. √ If d = 2 , and | [ u · τ ] | > 2 2 c ( ρ ) the linearized equations satisfy the weak Lopatinskii condition ⇒ weak stability, � p ′ ( ρ ) is the sound speed and τ a tangential unit where c ( ρ ) := vector to Σ . P. Secchi (Brescia University) Compressible vortex sheets
Introduction Compressible vortex sheets Compressible vortex sheets Linear Spectral Stability Main result Formulation of the problem Related problems Plan 1 Introduction Euler’s equations of isentropic gas dynamics Smooth and piecewise smooth solutions Existence results 2 Compressible vortex sheets Compressible vortex sheets Linear Spectral Stability Formulation of the problem 3 Main result Linear stability: L 2 estimate Linear stability: Tame estimate in Sobolev norm Nonlinear stability: Nash-Moser iteration 4 Related problems Weakly stable shock waves Subsonic phase transitions P. Secchi (Brescia University) Compressible vortex sheets
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